Questions tagged [wavefunction]
A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state. DO NOT USE THIS TAG for classical waves.
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Interpreting group velocity of free particle wave packet
I am trying to understand the concept of group velocity of a free particle wave packet: $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty} \phi(k)e^{ikx}e^{-\frac{i \hbar k^2 t}{2m}}dk.$$
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What's the lowest nuclear charge $Z < 1$ that will support a bound two-electron ion $(Z,2e^-)$?
In my programming project I calculate the minimal energy of an atom with 2 electrons in the $L=0, S=0$ state, using a Hylleraas wave function.
The values I find for $Z=2$ (He) and $Z=1$ (H$^-$) are ...
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What processes create or destroy information?
From a classical standpoint, it seems pretty clear that information can be easily lost. If you knock over a bookshelf and the books fall out, it seems like their initial order on the shelf cannot be ...
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Connection between singlet, triplet two-electron states and the Slater determinant
I'm confused about a number of things concerning two-electron systems and spin.
Here is (perhaps too much) exposition, skip to "the problem" if you want:
Consider the helium atom in the simplified ...
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Can particle quantum spin be described with a wave function? [closed]
I'm a little confused about the idea of spin. It's been non-technically described to me as "like magnetic dipole moment", except only two possible "directions". But I feel like that's a bad analogy, ...
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Parts of the Quark Wavefunction
Quarks are fermions meaning that they have an antisymmetric wavefunction. Under particle exchange the sign of the wavefunction.
The wavefunction is made up of a few different parts
$$
\psi_{Total} = \...
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Is the Wave Function a Unitary Operator? [closed]
A unitary operator can be represented as an exponential $$e^{iA}$$
and as we represent the wave function in general as $$e^{i k x}.$$ Does that mean that the wavefunction is unitary as the exponent is ...
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Confusion of Schrödinger equation and complex conjugates
I have a similar question that was asked in the following link:
(Schrödinger's Equation and its complex conjugate). But I find both the question and answers not specific enough.
So let me ...
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Wavefunction Collapse
I believe my Lecturer and the textbook have contradicted one another. My lecturer gave the example that if the spatial part of the wavefunction of a particle is given by
$\psi(x) = c_1\psi_1(x) + c_2\...
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1D Scattering Phase Shift (Finite Well) - Unphysical?
I am calculating the phase shift from a 1-dimensional potential well. This seems extremely simple, but I am just getting so confused by it.
Let there be a potential well of depth $V_0$ and spatial ...
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Taking Measurements of Quantities in QM
I have a quick question relating to Annihilation and Creation operators, and in taking observables in general.
Let's say, for instance, that I prepare a particle so that I consider the projection of ...
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What do operations on single Qubits of Unfactorable Superpositions Do?
So suppose I have the following Quantum Circuit:
A
---- |Control| -----|Hadamard|----
B
---- |xxxxxxx|------------------------
Which is a 2 input Controlled Gate (applying some gate of two choices ...
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Interpretation of boundary conditions in time-independent Schrödinger equation
The time-independent Schrödinger equation:
$$\ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$$
is second order, so we should expect the solution to have two "degrees of freedom" which can ...
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Full solution of the quantum particle on a ring
In quantum particle in a ring problem, the general solution for the wavefunction, with $k = R \sqrt{2 m E / \hbar^2}$, $R$ being the ring radius, $c_{+, -}$ being constants, $E$ the energy, and $m$ ...
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Tensor product in quantum mechanics?
I often see many-body systems in QM represented in terms of a tensor products of the individual wave functions. Like, given two wave functions with basis vectors $|A\rangle$ and $|B\rangle$, belonging ...
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Allowed Wave Functions of System
Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given ...
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How to find the minimum value of potential in QM?
In MIT problem sets I followed a solution of an exercise which focuses on odd-parity energy eigenstates in finite square well.
The point of problem is how to know or find the minimal value of ...
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Same quantum states represented in different basis
In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose and then ...
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Reflection in Finite Square Wells
For a Finite Square Well where we have a wavefunction $\psi(x)$ which is an energy eigenfunction with eigenvalue $E = 2V_0$ in the following potential:
$V(x) =
\begin{array}{ll}
6V_0 &...
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Nodes for a hydrogen atom probability density?
It is said that the wave function $\psi_{n,\ell,m}$ has $n-1$ nodes; $n-\ell-1$ from the radial part of the wavefunction and $\ell$ from the angular part. However, the probability density of finding a ...
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Wavefunction of a system of particles
A three-dimensional volume $V$ contains a certain number $N$ of electrons and they can't escape the volume $V$. Assume for simplicity that the potential $\mathcal{V}(\mathbf{r})$ is zero in all the ...
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Linear Combinations of Energy Eigenfunctions in 1D
Given that a particle is in a state defined by the wavefunction:
$$\Psi (x,t) = \psi_0(x)e^{-iE_0t/\hbar}+\psi_1(x)e^{-iE_1t/\hbar}$$ where $\psi_0(x)$ and $\psi_1(x)$ are the energy eigenfunctions of ...
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What is a wave function in simple language?
In my textbook it is given that
'The wave function describes the position and state of the electron and its square gives the probability density of electrons.'
Can someone give me a very simple ...
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What do the wave functions associated to the Fock states of each mode of a bound state system mean?
$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$
Consider a string of length $L$ under tension and clamped on each end.
This system is described by the wave equation and has a set of modes.
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Position and momentum measurement effects on wave functions
I have a few short questions about an interpretation of what happens with position and momentum wave functions described in literature I am using. Given momentum space wave function and position space ...
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Can there be a wave function that is physically possible but is non differerentiable (maybe even non-continous)?
The definition of a wave function demands continuity and differentiability so that it can satisfy the Schrödinger Equation. My question is whether this assumption is necessary for reality. Does ...
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Laughlin wave function and CFT
I have a question regarding Eq. (3.5) in Moore & Read's paper. They said
\begin{equation}
\Psi_{\text{Laughlin}}=\left\langle\prod_{i=1}^{N}e^{i\sqrt{q}\phi(z_i)}\exp\left[-i\int \mathrm d^2z^{\...
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Would a collection of entangled particles behave like a superfluid? [closed]
Superfluidity of a Bose-Einstein condensate comes from the fact that all the particles are found in the same quantum state. They are described by the same macroscopic wavefunction. They never collide ...
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Correlating two definitions of bound states in quantum mechanics
In Griffiths, he defines a bound state to be that stationary state for which the total energy E is such that $E<V(\pm\infty)$. Let $\psi(x)$ is a stationary state satisfying $E<V(\pm\infty)$ and ...
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Observer in the double slit experiment with photons
In the double slit experiment with photons, the interacting observer is an instrument, detector…
If you replace the detector with a piece of metal with the same mass as the mass of the detector, the ...
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Wave function of particle and antiparticle
The wave functions of particle and antiparticle are related by complex conjugation and wavefunction $Ψ$ must be complex for particle such as $n$, $p$. Is there way to prove this mathematically? Can we ...
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How to calculate probability of complex wave functions? [closed]
An election has an equation as such: $$Ψ(x) = e^{iαx^2}.$$
How am I supposed to find the probability of finding the electron over a certain range? Is Fourier Transform involved in this?
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Is there a way to prove that a bound state wavefunction can always be chosen real for an arbitrary potential in Quantum Mechanics?
As we can prove many things that always (at least in introductory quantum mechanical problems) apply using an arbitrary potential (like that $E>V_{\rm min}$ or else the solutions are non-...
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Quantum Mechanics: Can the probability of finding a particle in the whole space be smaller or higher at certain times?
In the book Introduction to Quantum Mechanics (by David Griffith) there is an Example 2.1:
Suppose a particle starts out in a linear combination of just two stationary states:
$$\Psi(x,0)~=~c_1\...
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Gauge transformation of wave function of a system of stationary charges
Let's say we have a system of $n$ stationary charges interacting via Coulomb potential. Let's ignore possible external electromagnetic fields. Moreover the system is quantum, and its wave function is $...
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Why does exchanging coordinates produce a phase of $\pm 1$ in an identical particle wavefunction?
Consider a system of two identical particles described by a wavefunction $\psi(x_1, x_2)$. There are two kinds of exchange operators one can define:
Physical exchange $P$, i.e. swap the positions of ...
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Particle here at a given time, in another galaxy a second later... Really?
I read "The Quantum Universe (Cox & Forshaw)" that a particle can be measured at a given position at a given time, and in another galaxy one second later. The probability of such event may be ...
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Three particles case, finding ground energy state
Here I came up with three particles in a box problem.
(Assumption: Here I do not consider the interaction between particles and spin for simplicity.)
What I want to do is express the ground state's ...
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Representing an ISW wavefunction graphically [closed]
I'm trying to decode this diagram given to us for an assignment.
The description of the diagram is 'Consider a particle of mass m confined to a 1-dimensional square well, given graphically by
the ...
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What is a linear polarized photon?
According to Dirac a 'linear' polarized photon is a superposition of left and right rotating photons. Here is a puzzling aspect of this superposition.
There are dichroic materials which can absorb ...
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Calculating the energy of a particle using the Time Independent Schrodinger Equation [closed]
If we have a wave function $\Psi(x,t=0)$ which is a solution to the TISE for a zero potential in an infinite square well, would calculating the energy at $t = 0$ at a position be as easy as ...
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Exchange principle in terms of states and coordinates?
I have seen the exchange principle written in two ways, one in terms of coordinates and the other in terms of states:
If $\psi_{AB}(1,2)$ represents particle $A$ in state $1$ and particle $B$ in ...
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Meaning of the Vector Wave Equation
So I thought I would try my luck here on physics stack exchange about an intuitive meaning of the Vector Wave Equation. I know there are a lot of resources out there that explain this equation, but ...
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Significance of $i$ in the Schrödinger equation [duplicate]
There's an imaginary $i$ in the Schrödinger equation, which I guess is to define the position of the particle in a space-time involving a complex function. But what is the real physical significance ...
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1D transmission lines wave equation solution
you may know that the solution of 1D wave equation by d’Alembert is F(x-ct)+F(x+ct) and my question is that like is this F(x-ct) at transmission lines only the equation of one forward going wave that ...
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The Hilbert space that contains the first order correction to the state vector in Time-independent Perturbation Theory
When deriving the expression for the first order correction to the state vector of the new hamiltonian ( $H = H^0 + H^\prime$ )
we assume that
$$|\psi_n^1\rangle = \sum_{m ~\neq~ n} C_m^{(n)}|\psi ...
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Wave function for step potential
Given the step potential
$$V(x)=\begin{cases}
0~~~~~~~~\text{if }~~x \leq 0 \\
V_0~~~~~~\text{if }~~x > 0
\end{cases}$$
Consider the case where $E < V_0$. In this region $x \leq 0$ we have ...
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Help normalising and taking the inverse Fourier transform of this wavefunction [closed]
Normalising
Consider the wavefunction $$\psi(x,0)=Ne^{-\frac{|x|}{\lambda}}.$$ In order to normalise this I take the integral, which due to the modulus on the $x$ I evaluate just from zero to ...
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Connection between Hamiltonian version of the least action principle and probability amplitude in the Schrödinger equation
If I'm not mistaken, Schrödinger was influenced to look at wave equations because of de Broglie's assertion about particles having a wavelength. He started with the Hamiltonian equation which is ...
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Quantum Mechanics: How to compute how fast must a function go to zero at infinity? [closed]
We say that the wave function must go to zero at infinity faster than $1/x^{0.5}$ in order for it to be normalizable.
What about other quantities like the probability current? What is the general rule ...